In the art of robotics, the term generalized coordinates refers to a characterization of the system that uniquely defines its configuration. For example, in a two-link planar manipulator, the robot has two degrees-of-freedom (DOF) so two joint variables, θ1 and θ2, can be referred to as generalized coordinates as these variables and may be used to define the position of the robot. Thus, control of the position of the end-effector of the manipulator may be done in the joint space since with the knowledge of each joint angle, the position of the end-effector may be determined.
In practice, it is often the case that the desired position of the end-effector is given in terms of an operation space also known to those of skill in the art as a task space. For the two-link planar robotic manipulator an example of an operation space is the Cartesian coordinate system defining a plane of operation of the robot. An operator of a robot may require that the end-effector be moved to a specific point, P(x, y), in order to perform a task, such as placing a spot weld, or picking a part on an assembly line. Defining the position of the robot in the Cartesian coordinate system is advantageous for an operator of the robot. However, a description of the point P(x, y) does not uniquely define the configuration of a robotic arm in terms of the joint variables. As is understood, a two-link planar manipulator can reach the same point P(x, y) using two different configurations: an elbow-up configuration or an elbow-down configuration. A manipulator with redundant degrees-of-freedom may have more than two configurations that allow the manipulator to reach the same point in an operation space further complicating the choice of a robot joint configuration that should be used to perform a task. Because an operation space may not uniquely define a configuration of a robotic arm, joint space control, i.e. control of the robot joint variables θ1 and θ2, cannot be readily performed. This is because it may not be obvious how to specify a joint space trajectory for each of the manipulator's joints to control the position of the end-effector.
A fundamental challenge associated with operation space control as described is the fact that the rows of the Jacobian J(q) may lose rank (become singular). In this case, it is not possible to compute a control vector in the joint space that produces the necessary operation space forces. A two-link planar manipulator has a singular Jacobian when θ2=0 and when θ2=π. In these cases, the two links of the manipulator are co-linear and forces cannot be produced in the degenerate direction. This implies that there is no combination of joint torques, τθ, that can produce a desired force vector, Fx. One approach for circumventing the problem is to detect the potential for singularities by identifying degenerate directions and setting the commended force in those directions equal to zero. Generally, by identifying degenerate directions appropriately, it is possible to compute control parameters such as a control torque vector that allows the manipulator to pass through the singular configuration and complete a desired task. However, typically when the manipulator operates near a singular configuration, the values of control parameters such as computed control torques can change sharply and suddenly. These sudden changes in control parameters can excite vibrational modes in manipulator links and potentially reduce the accuracy of the robotic control system. In addition, suddenly changing the control torque command may be damaging to the joint actuators. It is therefore advantageous to develop control methodologies that avoid sudden and sharp changes in the manipulator control torques.
Disclosed here is a means for mitigating this issue by generating control parameters based using a plurality of distributed joint angles having a distribution generally about a given joint angle in a robotic manipulator. The disclosed controller determines and communicates control parameters to at least one actuator in order to generate motion of the robotic arm and displacement of the end effector in a manner that mitigates the sharp and sudden changes in commanded control parameters associated with the currently utilized methods. The method is particularly useful when applied in the neighborhood of joint angles generating a singular Jacobian.
These and other objects, aspects, and advantages of the present disclosure will become better understood with reference to the accompanying description and claims.